Enumerating pattern-avoiding permutations by leading terms

TL;DR

This paper provides a direct counting proof for the enumeration of pattern-avoiding permutations with fixed leading terms, extends the analysis to fixed prefixes, and classifies Wilf-equivalence classes for patterns of length three.

Contribution

It offers a new proof method, generalizes enumeration to fixed prefixes, and classifies Wilf-equivalence classes for classical and vincular patterns of length three.

Findings

Enumeration formulas depend on prefix length, extrema, and order statistics.

Exact counts for permutations avoiding specific patterns with fixed prefixes.

Classification of Wilf-equivalence classes for patterns of length three.

Abstract

The number of 123-avoiding permutation on $\{1,2,\ldots,n\}$ with a fixed leading terms is counted by the ballot numbers. The same holds for $132$-avoiding permutations. These results were proved by Miner and Pak using the Robinson-Schensted-Knuth (RSK) correspondence to connect permutations with Dyck paths. In this paper, we first provide an alternate proof of these enumeration results via a direct counting argument. We then study the number of pattern-avoiding permutations with a fixed prefix of length $t\geq1$, generalizing the $t=1$ case. We find exact expressions for single and pairs of patterns of length three as well as the pair $3412$ and $3421$. These expressions depend on $t$, the extrema, and the order statistics. We also define $r$-Wilf equivalence for permutations with a single fixed leading term $r$, and classify the $r$-Wilf-equivalence classes for both classical and vincular patterns of length three.